NASA Technical Memorandum107691
MODELING AND MODEL SIMPLIFICATION
OF AEROELASTIC VEHICLES :
AN OVERVIEW
Martin R. Waszak and Carey S. Buttrill
Langley Research Center
Hampton, Virginia
David K. Schmidt
Arizona State Universtiy
Tempe, Arizona
September 1992
National Aeronautics and
Space Administration
Langley Research CenterHampton, VA 23665
(_..ASA T_4-1076el) MODELING AND
:_!]OFL 5INPLI61C,aTION OF AEROELASTIC
V_HICLES: AN OVERVIEW (NASA) 20 p
N93-12216
Uncl as
G3/08 01.29288
https://ntrs.nasa.gov/search.jsp?R=19930003028 2018-07-29T09:38:01+00:00Z
Modeling and Model Simplification
of Aeroelastic Vehicles: An Overview
Abstract
The rigid-body degrees of freedom and elastic degrees of freedom of aeroelastic vehicles are typically
treated separately in dynamic analysis. Such a decoupling, however, is not always justified and modeling
assumptions that imply decoupling must be used with caution. The frequency separation between the rigid-
body and elastic degrees of freedom for advanced aircraft may no longer be sufficient to permit the typical
treatment of the vehicle dynamics. Integrated, elastic vehicle models must be developed initially and
simplified in a manner appropriate to and consistent with the intended application. This paper summarizes
key results from past research aimed at developing and implementing integrated aeroelastic vehicle models
for flight controls analysis and design. Three major areas will be addressed; 1) the accurate representation
of the dynamics of aeroelastic vehicles, 2) properties of several model simplification methods and 3) the
importance of understanding the physics of the system being modeled and of having a model which exposes
the underlying physical causes for critical dynamic characteristics.
Introduction
The means of obtaining the simplest valid mathematical model of an aeroelastic vehicle for dynamic
analysis and control system design is a major issue in flight vehicle dynamics. The need to account for
aeroelastic effects will make model formulation very important for flight vehicles of the future. Reduced
structural weight, potential for static instability, and application of high-authority feedback control systems
will result in reduced frequency separation between the "rigid-body" modes and "elastic" modes. In
addition, the potential for using control systems to influence the vehicle configuration, the so-called control
configured vehicle concept [Schwanz (1977)], will require accurate aeroelastic models to be available very
early in the design cycle.
Of particular importance is the potential for dynamic aeroelastic effects to influence "rigid-body" vehicle
responses. Schmidt (1985), and Swaim and Poopaka (1982) have addressed the effects of aeroelastic/rigid-
body modal coupling on flying qualities. A simulation study using the elastic vehicle model from Waszak
and Schmidt (1988) and performed in the Visual and Motion Simulator (VMS) facility at NASA Langley
Research Center addressed these effects [Waszak, Davidson, and Schmidt (1987)]. This study showed that
increasing structural flexibility, even to moderate levels, can have a negative impact on vehicle handling
qualities.
The view of the authors is that an integrated elastic vehicle model should be developed initially and
simplified in a manner consistent with the intended application. This must be done in such a manner that the
salient dynamic effects are retained in the model. This view must also be tempered with the need to have the
simplest model possible to facilitate effective dynamic analysis and control synthesis, and to ease
computational burden.
MR. Vvaszaka_M C.S. Buttrill
In this paper, several key aspects of formulating aeroelastic models for flight dynamics applications will
be addressed. The development of the equations of motion for any aeroelastic vehicle is presented in which
the structure is in many ways very similar to the equations of motion for a rigid vehicle. The similarities and
differences of the aeroelastic equations of motion and traditional rigid-body equations of motion will be
discussed and the important features of the aeroelastic model structure will be addressed.
The methods used for model simplification depend to a large degree on the ultimate use for the reduced-
order model. However, one must never lose sight of the objective: obtain the simplest vehicle model that
possesses the requisite accuracy. Techniques capable of delivering valid reduced-order models for control
system design will be discussed.
Another key issue associated with the inherent complexity of aeroelastic vehicles is interpreting vehicle
behavior. Understanding the sources of undesirable (or desirable) dynamic behavior is often required to
design control systems or to design the airframe itself. The structure of the equations of motion and
properties of the model simplification methods can aid the analyst/designer in developing this
understanding. This is an area which is often overlooked in the development of modeling and model
simplification methods and will be specifically addressed herein.
Equations of Motion
The development of the equations of motion of an elastic aircraft has been addressed many times in the
literature [Milne (1962), Roger (1977), Schwanz (1977), Cerra and Noll (1986)]. This subject has recently
been revisited by the authors with emphasis on the need to develop accurate aeroelastic vehicle models and
to clarify the assumptions associated with their development and assess their validity. The development of
the equations of motion discussed here are intended for application to flight dynamics, simulation and
control system design for elastic aircraft. This application places special requirements on the form and
properties of the resulting equations. They must be able to describe large amplitude maneuvers in a body
reference coordinate system while simultaneously describing the small amplitude structural deflections.
This requires that the model be nonlinear in variables which describe body orientation, but allows the
structural dynamics to be linear. The equations must also account for inertial and aerodynamic coupling
which are normally neglected. An additional goal is to have the form of the equations be applicable over the
entire design cycle, from the conceptual through detailed design phases.
Two recent unrelated studies, one by Waszak and Schmidt (1988) and another by Buttrill, Zeiler, and
Arbuckle (1987), have sought to develop the equations of motion for elastic aircraft to meet these goals.
The emphasis of the first study, which will be referred to as the Waszak Study, was the assembly of a
mathematical model which integrated "rigid-body" and "elastic" degrees of freedom with particular
emphasis on the assumptions made at the various stages in the development and on obtaining a set of
equations that constitute an analytical (or literal) model. The second study, which will be referred to as the
Buttrill Study, focused on including effects of nonlinear inertial coupling between rigid-body angular rates
and structural deformations and rates which are usually ignored in conventional aircraft modeling.
Modeling and Model Simplification of Aeroelastic Vehicles
Both studies utilized Lagrange's method of deriving the equations of motion relative to a "mean axis"
body reference coordinate system. The use of mean axes minimizes the degree of inertial coupling between
the rigid-body and the elastic degrees of freedom. As a result of using mean reference axes and some key
simplifying assumptions, the equations of motion from the Waszak Study have a structure similar to
conventional rigid aircraft (with additional degrees of freedom associated with the elastic modes). These
equations have the added feature that they exhibit no inertial coupling between the elastic and the rigid
degrees of freedom (all coupling occurs through the aerodynamic forces). The Buttrill Study sacrificed
some of the similarity to rigid aircraft demonstrated in the Waszak Study in the interest of more accurately
modeling the inertial coupling effects. This results in the body axis rotational equations having additional
coupling terms.
Both studies also made use of the same basic assumptions in the derivation of the equations of motion;
a) the structure is treated as a collection of lumped masses with constant mass density,
b) the structural deformations are small (i.e. linear stress-strain and linear stain-displacement relations
are valid),
c) the structure exhibits synchronous elastic motion described by a complete set of normal modes, and
d) a local Earth-fixed inertial reference frame with constant uniform gravitational field was utilized.
The two studies differ slightly in that the Waszak Study used the additional assumptions that;
e) each mass element is a point mass with no rotational inertia, and
f) elastic deformation and rate are sufficiently small or colinear so that their cross product is negligible
and the inertia tensor is constant.
The equations of motion from the Buttrill study can be simplified to a form identical to those in the
Waszak study by adding these assumptions, therefore the two derivations are completely consistent.
Mean Axes
A short discussion of mean axes is warranted before addressing the specific issues associated with the
two studies. As previously stated, the form of the equations of motion is facilitated by the use of a mean
axes reference frame. The mean axis reference frame is not fixed to a material point in the body but floats
so that its origin is always at the instantaneous center of mass of the body. Furthermore, the mean axis
frame is oriented in such a way that the reference frame motions are inertially decoupled from the structural
deflections. An excellent survey of five types of floating reference frames is given in Canavin and Likins
(1977). The practical mean axis conditions of Waszak and Schmidt (1988) are equivalent to the Buckins or
linearized Tisserand frame of Canavin and Likins (1977).
Elastic modes of free vibration calculated from a structural model unconstrained in translation and
rotation should satisfy the practical mean axis conditions [Canavin and Likins (1977)]. If the structural
model has been restrained to create a nonsingular stiffness matrix, the resulting modes are inertially coupled
with motion of the body frame and the equations of motion become more complex.
3
MJL Waszak and C.S. Buttrill
In the remainder of this section the important results from each of the two studies will be reviewed.
Note that the equations of motion discussed herein are in a form which is applicable to a wide class of
problems. Any airplane which has significant elastic dynamics and needs to be described dynamically in a
body reference frame can be represented using these equations.
The Waszak Study
Tables 1 and 2 summarize the form of the equations of motion of an arbitrary elastic aircraft derived in
Waszak and Schmidt (1988). Table 1 contains three translational equations and three rotational equations
that describe the motion of the body reference coordinate system and a set of equations which describe the
structural deformations relative to body axes.
Notice that the translational and rotational equations are identical in form to conventional rigid-body
equations of motion. The differences lie in the representations of the aerodynamic forcing functions. These
forces completely describe the coupling which exists between the rigid-body and elastic degrees of freedom.
Also note that the moments of inertia, I(.), correspond to the deformed structure and change as the vehicle
deforms. It is assumed, however, that the variations in the moments of inertia due to elastic deformation are
small and are hence neglected. The moments of inertia actually used in the equations of motion correspond
to the undeformed vehicle.
In addition to the body axis translational and rotational equations there are a set of equations that
describe the elastic deformations of the body. The elastic mode equations are typical of a second-order
oscillator with equivalent modal damping proportional to modal velocity. The only coupling with the rigid-
body degrees of freedom is through the generalized aerodynamic forces, QJn (see Tables 1 and 2).
Table 2 summarizes expressions for the aerodynamic forces for an elastic aircraft. These expressions are
presented in a stability derivative form. The difference between these expressions and those used for rigid
aircraft is in the addition of elastic stability derivatives. These terms are similar to their rigid aircraft
counterparts and serve to couple the elastic degrees of freedom with the rigid-body degrees of freedom. The
relationships between the structural parameters (i.e. mode shapes) and the elastic stability derivatives are
developed in Waszak and Schmidt (1988) by using strip theory aerodynamics. The validity of this
Table I - Elastic aircraft equations of motion
M [ 13 - rV + qW + g sin0 ] = Qx
M [ _/- pW + rU - g sin_ cos0 ] = Q¥
M [ "_V- qU + pV - g cost_ cos0 ] = Qz
Ixxt_ - (IxyCt + Ixzr ) + (Izz - Iyy )qr + (Ixyr - Ixzq )p + (r 2 2-q )Iyz = Q_
Iyy_t -(Ixyl_ + Iyzr ) + (Ixx - Izz )pr + (Iyzp - Ixyr )q + (p2 _ r2)ixz = Qo
Ixxr- (Ix_ + Iyzq ) + (Iyy- Ixx )pq + (Ixzq - Iyzp )r + (q2 _ p2)ixy = QV
Mj[ 'iij + 2;jt.oj(I + _jzrlj ] = Qq ; j=1,2,3 ....
4
Modeling and Model Simplification of Aeroelastic Vehicles
Table 2 - Elastic aircraft generalized forces
Qx-
oo oo
pVdS -_--( Cx_l_ + Cxqq + Z CxilTli2 (Cxo+Cxtt_+Cxa_+ ZCxrlTlii ) +P Sc i. )+Txi=l i=l
pVgS oo 9VoS b ooQY - 2 (Cy0+CYI3[_+CYa_+Z i )+--Z i"Cyrllli 4 CyilTli + Ty
i=l i=l
Qz-
PV02S oo2 ( Cz0 + CZ_,cx + Cza _5+ _
i=l
oo
Cz_,qi) + (%1_ + Czpp + Czqq + Z i. ) +Tzi=l
PVo2SbQ00 - 2 ( CLo
Qo -
Q_-
oo PVoSb 2 ooi )+ (%P+ +Z i+ CLt_[$+ CLa_+ _ Cunrli 4 Cuqq Cu; ) + br
i=l i=1
PV02S_ oo PVoS_2 oo2 ( CM0 + CM_ 0_+ CMa8 + _ CMllrlii ) + 4 ( CMdc_ + CMq q + _ CM_"ill ) + MT
i=l i=l
PVoSb 2 ,,oPv0Zsb i ) +-- ( CNpP + CNrr + Z CN_lqii" ) + NT2 ( CN0 + CNI_ + CNa_+ Z CNrlTli 4
i=l i=l
0vgse2 c5, E cJ n )+i=l
oo
pV°Sc2 CJpp + CJqq + C_r + Z .i.4 ( Cj_ + " cl_ffli )i=l
approximation will not be discussed further other than to say that it is a reasonable approach for high aspect
ratio configurations if numerical values are sought. However, the real importance is that the expressions for
the elastic stability derivatives obtained using strip theory provide physical insight through a conceptual link
between physical parameters and their effect on the equations of motion independent of their numerical
values.
The use of stability derivatives has the added advantage that the same form of the equations of motion
can be used throughout the design cycle. Early on, before detailed analyses have been performed, strip
theory or other first-order methods can be used to obtain a preliminary elastic vehicle model. Later, when
more detailed analyses have been performed (e.g. computational fluids dynamics analyses, wind tunnel tests,
finite element analyses), the data can be converted to a stability derivative form and substituted directly into
the equations of motion.
The equations of motion summarized in Tables 1 and 2 also constitute a "literal" model for an elastic
airplane. The literal model can be used to develop insight into the effects of various physical parameters on
the dynamic characteristics of the vehicle. Such insight is difficult to obtain from purely numerical models.
It is important to note that even if a "numerical" model is available, it can be put in a form consistent with
the "literal" model which allows the analyst to exploit his insight.
An example of the importance of modeling aeroelastic dynamics was demonstrated in Waszak and
Schmidt (1988). The equations of motion discussed above were applied to a high speed transport aircraft
with a moderate level of structural flexibility. The results of the study showed that neglecting aeroelastic
5
M.R. Waszak and C.S. Buttrill
dynamics during model development resulted in a model which incorrectly indicated the vehicle to have a
stable phugoid mode and which had errors in short period frequency and damping of approximately 55%
and 14%, respectively, compared to the complete aeroelastic vehicle model.
The Buttrill Study
The equations of motion developed during the Buttrill Study [Buttfill, Zeiler, and Arbuckle (1987)] were
refined by Zeiler and Buttrill (1988) and are summarized in Table 3. Zeiler and Buttrill (1988) utilized
nonlinear strain-displacement relations to improve the calculation of [A2j]jk , which appears as a stiffness
term when nonzero rotational rates, co, are present. These three sets of equations are presented in vector
form and correspond to the scalar equations presented in Table 1 except for the addition of terms
representing nonlinear inertial coupling. In Table 3, an open dot over a quantity indicates the time rate of
change of that quantity expressed in body frame components.
The translational equation is completely analogous to the previous study. The use of mean axes has
eliminated any inertial coupling between the rigid-body degrees of freedom and the elastic degrees of
freedom. The rotational equation, on the other hand, has additional terms not included in the Waszak Study.
These additional terms account for variations in the inertia terms with structural deformation (deflection
induced changes in mass distribution) and second-order coupling associated with the fact that
the cross product between structural displacement and rate are nonzero (since all modes do not act in the
same plane). When these terms are neglected the rotational equations are completely analogous to those in
the Waszak Study.
The elastic mode equations also have additional terms. These are associated with angular acceleration of
the body reference frame, Coriolis acceleration, and centrifugal loading. Neglecting these effects also
simplifies the elastic mode equations to those from the Waszak Study.
The objective of the Buttrill Study was to generate a high fidelity model of an elastic airplane with
special attention to the effects of inertial coupling. Consequently, the representation of the aerodynamic
forces was not explicitly addressed other than to discuss the numerical methods by which aerodynamic
forces were computed for an example problem. It should be noted, however, that coupling between the
rigid-body and elastic degrees of freedom also occurs through the aerodynamic forces. The terms F, L, and
Q_j are applied loads and include aerodynamic and thrust forces and moments.
Table 3 - Elastic aircraft equations of motion with inertial coupling
m_ = F-m(co x V)+mg
[J]_o+h_jklqJrl'k = L +cox [J]¢o- [_]_-hjkrlJrlk-_×hjkrlJrl k
"'k o k 20).hjkl_k 1 T [A2j]jkl] k }coMjkT ! -_.hjkT I = Qrlj-Mjjcoj2TIj+ __ +_CO {[AJ]j+
[J] = [Jo] + [ AJ ]jrl j + [ A2j ]jkrlJrl k
6
Modeling and Model Simplification of Aeroelastic Vehicles
In summary, it was found that in general, nonlinear inertial coupling can become a significant if at least
one of the following characteristics are reflected in the vehicle:
a) aerodynamic loads are small,
b) expected rotational rates are of the order of the elastic frequencies,
c) the model geometry is sufficiently complex that transverse deflections result in changes in mass
distribution.
Summary
Both of these studies indicate that the structure of the equations of motion of elastic aircraft are quite
complex, even when they are developed with the intent of minimizing the apparent modal coupling. Some
of the complexities associated with the dynamics of aeroelastic vehicles are listed below.
a) The models which result are of high dynamic order, two additional states for each elastic mode.
b) The relationships between the various model parameters that are fairly well understood for rigid
2 ZwMq-M_) are more difficult to identify for elastic aircraft.aircraft, (e.g. O)sp =
c) The parameters that appear in the model, such as the generalized modal stability derivatives, are less
well understood than classical parameters such as La, and accurate numerical values are more
difficult to obtain.
d) There are significant uncertainties associated with the elastic parameters.
The complexity negatively impacts many aspects of aeroelastic aircraft dynamics and control. A natural
next step is to address how the important effects of elastic dynamics can be retained in the model while at
the same time simplifying the model structure. This is discussed in the next section.
Model Simplification
The design of effective and practical control systems requires that the designer understand which aspects
of the vehicle dynamics are important, the uncertainties associated with the model, and a knowledge of the
parameters which have a significant impact on critical vehicle responses. The size and complexity of the
models which result from describing the aeroelastic interactions (via the equations of motion discussed
previously) make this an extremely difficult task. It is likely, however, that once the key interactions have
been accounted for, many fewer physical parameters need to be retained to capture the prominent aspects of
the vehicle responses. It is therefore important to be able to simplify the models but still retain enough
information to capture the salient features of the aeroelastic interactions. Note that the parameters which
turn out to be key are not usually discemable before the detailed model is obtained and requires that such a
model be obtained first followed by appropriate simplification.
Model simplification is also important for aeroelastic systems from the perspective of controller
complexity. Many control design methodologies require the system model to be linear and result in
controllers which are of dynamic order equal to or greater than the design model. The size and complexity
7
M.R. Waszak and C.S. Buttrill
of aeroelastic models therefore dictates large, complex controllers. A reduced-order linear model which
retains the salient aspects of the nonlinear system dynamics within a simplified form may allow effective
controllers to be designed with significantly simpler structure.
Model simplification has as its goal to obtain a model which, while simpler than the full-order model,
approximates some aspect of the true system. The first step in this process is to linearize the system
dynamics about an equilibrium condition. Reduction of the linear model is then performed and the desired
manner in which the reduced-order linear model approximates the full-order model depends to a large
degree on the intended application. For example, in control system synthesis it is important to accurately
represent the system frequency response in the frequency range where crossover of the loop transfer function
is likely to occur [McRuer, Ashkenas, and Graham (1973)]. Note that this implies that there are frequencies
both above and below the critical frequency range which may not need to be well modeled. The frequency
range of interest is very important when applying model simplification and plays a key role.
There are many methods by which linear elastic aircraft models can be simplified. A few of these will
be discussed herein and are summarized below.
1. Truncation - deletes some of the modes (modal truncation) or states (state truncation) from the full-
order model [Waszak and Schmidt (1988)].
2. Residualization - accounts only for the static effects of some modes or states whose dynamics are not
crucial [Kokotovich, O'Malley, and S annuti (1976)].
3. Balanced reduction - minimizes frequency response error in a normed sense and has certain
advantages associated with obtaining desired accuracy which will be addressed subsequently [Bacon
and Schmidt (1989)].
4. Literal (or symbolic) simplification - addresses the impact of various physical parameters on the
system responses and ignores those which have little impact [Schmidt and Newman (1988)].
Each of these methods have advantages and disadvantages as they apply to simplification of elastic
aircraft models. In this section the four simplification methods will be discussed in these terms.
Truncation
Truncation is a common form of model reduction. In fact, it is the most common form of reduction since
every finite dimensional linear model is a truncated model in the sense that there is always some aspect of
the physical system that is neglected in the modeling process. While truncation is a well established model
simplification technique, a slightly different view based on Cramer's Rule is presented here with some
interesting implications [Schmidt and Newman (1988)].
The degree to which truncation can be utilized depends on the degree to which the truncated degrees of
freedom directly influence the vehicle response and the degree to which they couple with the retained
degrees of freedom. Consider a frequency domain representation of a linear system as shown below.
Modeling and Model Simplification of Aeroelastic Vehicles
l[z s 1r(s) m(s) Zr(S) = br(S) U(s)
Y(s) = M(s) Z(s) + mr(S) Zr(S)
(la)
(lb)
Y(s) is the vector of responses, U(s) is the vector of commands, and [ZT(s), Zr(S)] T is the vector of states or
system degrees of freedom. Assume for simplicity of discussion that Zr(S) is a scalar. In this case m(s) is a
scalar, r(s) and br(S ) are row vectors, and c(s) and mr(S ) are column vectors. By applying Cramer's Rule for
the determinant of a matrix [Strang (1980)], and an identity for the determinant of a partitioned matrix
[Brogan (1974)] the transfer functions for the ith output due to the jth input of the system can be written -
Zi(s ) det{ (Ai(s)]Bi(s))-c(s)m-l(s)(ri(s)[bri(S))}
Uj(s) - det{ A(s)- c(s)m-l(s)r(s) }(2a)
Zr(S) brj(S) det{ A(s)-Bi(S)bri-l(s)r(s) }
Uj(s) - m(s) det{ A(s)-c(s)m-l(s)r(s) }(2b)
where the notation A i IBj represents the operation of replacing the ith column of A with the jth column of B.
These forms of the system transfer functions are very useful in identifying some important aspects of
applying model reduction via truncation.
From inspection of the transfer function expressions above one can see that if Ck(S)rl(S ) << m(s) and
ck(s)(r i Ibrj)/(s) << m(s) ; k,l = 1,2 ..... n where n is the number of states, then
Zi(s) Zi.(s) det{ Ai(s)I Bi(s)}Uj(s) - Uj(s) - det{ A(s) }
(3)
Also, if in addition Bjk(S ) r/(s) << brj(S) ; k,l = 1,2 ..... n then
Zr(S) br'(_s)
Uj(s)- m(s)(4)
Equation (3) is exactly what results if the degree of freedom zr is truncated from the model (or not included
in the model from the outset).
Examining the transfer function zr(s----_)from the context of physical systems one finds that the polynomialUj(s)
m(s) is usually of higher order than brj(S). Therefore, for frequencies above some value determined by m(s)
and brj(S ) (as s---_) the effect of zr on the output Y(s) is negligible which can be approximated by
9
M.R. Waszak and C.S. Buttrill
A
Y(s) = M(s) Z(s) (5)
where the elements of Z(s) are described by
A
^ X ujZi(s ) = (s) (6/j J
The implication is that truncation should be used to simplify a model in which the degree of freedom to be
removed is much slower than the dynamics of interest.
The conditions that must be satisfied to apply truncation to slow dynamics have some important
implications. The condition that Ck(S)r/(s ) << m(s) implies that the degree of coupling between the deleted
of freedom and the retained degrees of freedom is small. Similarly, ck(s)(r i [brj)/(s)I
degree << m(s) implies
that the coupling effect between the deleted degree of freedom and the retained degrees of freedom (via c(s))
combined with the ability of the jth control input to excite the deleted degree of freedom is small. Finally,
Bjk(S ) r/(s) << brj(S ) implies that the combination of the coupling effect between the retained degrees of
freedom and the deleted degree of freedom (via r(s)) and the ability of the jth control input to excite the
retained degrees of freedom is small. These conditions will be referred to as the "decoupling conditions"
and can generally serve as a test to determine if a particular degree of freedom can be legitimately truncated.
Note that these conditions are identically satisfied if the system is in modal form since the off-diagonal
partitions, c(s) and r(s), are identically zero in this case. Therefore, modal truncation can be effectively
applied whenever there is sufficient frequency separation between the deleted mode and the dynamics of
interest, so that in the frequency range of interest mr(S)Zr(S) can be neglected from the output equation, Eqn.
(lb).
The truncation of slow modes is contrary to the typical use of truncation which is to remove higher order
dynamics. The reason that truncation works for some higher order dynamics can be seen by addressing the
problem by using a partial fraction expansion of a transfer function,
Y(s) R 1 R 2 R n-- +-- +... + _ (7)
U(s) - S+_l s+_2 S+_n
where R i are the residues and _'i are the eigenvalues of the system [D'Azzo and Hoopis (1975)]. If the
desired frequency range of accuracy of the simplified model is well below _n' then the last term in the
R n
expansion can be approximated by_--__. Since this term is associated with a high frequency mode, the value11
of Ln is most likely much greater than unity. In addition, high frequency modes are frequently difficult to
excite which results in small residues. Clearly, if R n << Xn then the last term in the partial fraction
expansion can be neglected without much impact on the frequency response in the frequency range of
interest. Thus, truncation can be used to remove both low and high frequency dynamics when the
appropriate conditions are satisfied.
10
Modeling and Model Simplification of Aeroelastic Vehicles
Residualization
Residualization is another common method of model simplification. Many times a system may have
some dynamics that are fast compared to the dynamics of interest [Kokotovic, et al (1976)]. However, the
fast dynamics can interact with the slower dynamics so that truncation of the fast dynamics may not be valid.
Residualization allows one to take into account the interaction without including the dynamic effects of the
fast dynamics.
The same model structure used previously in the discussion of truncation will be used again here.
Consider the frequency domain representation of a linear system presented in Eqns. (1). The system transfer
functions can again be approximated by the expressions in Eqns. (3) and (4) when the decoupling conditions
are satisfied (i.e. Ck(S)r/(s ) << m(s), Ck(S)(ri Ibrj)/(s) << m(s), and Bjk(S ) r/(s) << brj(S)).
Residualization is typically accomplished by letting the degrees of freedom to be removed from the
model reach their steady state values instantaneously by setting their derivatives zero. An analogous
_ Zr(S)interpretation is to let s-->0 in the transfer tunction _ (as opposed to letting s--->oo for truncation).
The simplified model using residualization takes on the following form.
Y(s) = M(s)Z(s) + mr(S) bri(0)U(S)m(0) (8)
The implication here is that residualization can be legitimately applied to degrees of freedom which are
much _faster than the retained modes. Therefore, only degrees of freedom whose frequencies are well above
the expected crossover frequency range should be considered for residualization.
Notice again that the decoupling conditions are automatically satisfied when the system is in modal
form. When the frequency range of interest is well below _'n, the simplified model produced by modal
residualization (Eqn. (8)) is identical to the partial fraction expansion of the transfer function (Eqn. (7)) withr}
the last term approximated by _n.11
While the validity of performing model simplification via truncation or residualization can be evaluated
by the degree to which the decoupling conditions are satisfied and the degree of frequency separation
between the deleted dynamics and the desired dynamics, there is no guarantee on the accuracy of the
resulting simplified model. The current approach is a cut and try (i.e. iterative), graphical procedure. A plot
of the candidate frequency response is compared to that of the full-order system and a decision is made as to
its acceptability. This is a basic limitation of these approaches.
An advantage of these approaches, however, is that the form of the model which results after
simplification is the same as the corresponding portion of the original. Therefore, if the model had a special
structure before simplification then that structure is retained in the simplified form. This can be important in
allowing the analyst to use his knowledge of the physics to interpret the accuracy of the resulting simplified
model as well as the effect of various physical parameters on the system response.
11
M.R. Waszak and C.S. Buttrill
A comparison of truncation and residualization applied to a high speed transport aircraft is presented in
Waszak and Schmidt (1988). The results indicate the these methods are often quite acceptable. However, as
structural flexibility increases these methods may not provide the required accuracy for the desired model
order.
Balanced Reduction
Internally balanced reduction and frequency weighted internally balanced reduction are two more model
simplification methods that have received considerable attention recently. A considerable body of literature
has addressed the concept of balanced reduction and its variants [Enns (1984), Bacon and Schmidt (1989),
Glover (1984)]. We will simply address some of the issues that should be kept in mind when considering
this model simplification approach.
Briefly stated, the balanced realization approach to model reduction chooses an ordered combination of
state directions which dominate the input/output behavior of the system in decreasing order. An advantage
of this approach is that the accuracy of the reduced-order model can be measured in a normed sense. The
frequency response error between the full-order system and the simplified system is bounded by twice the
sum of the Hankel singular values of the deleted degrees of freedom [Bacon and Schmidt (1989)]. In
addition, the measure of accuracy is a direct by-product of the reduction process.
Unfortunately, the balanced reduction method results in a simplified model which matches the frequency
response of the full-order model in regions where the magnitude is greatest. This may not be the region of
crossover. As a result the model may not be acceptable for application to control design regardless of the
"accuracy." An example of this limitation is presented in Schmidt and Newman (1989). The weakness of
the method can be resolved by applying weighting functions to the basic approach to emphasize a desired
frequency range (e.g. the crossover region) [Enns (1984), Bacon and Schmidt (1989)]. However, when this
is done the measure of the accuracy of the simplified model is no longer valid. Research into resolving this
issue is currently underway.
Yet another limitation of the balanced reduction methods is associated with the fact that the state space
form of the simplified model which results is only related to the original model through their frequency
responses. The states of the simplified model are entirely different from those of the full- order model. In
fact, the simplified state space model looses all structure that appeared in the full-order model. The
implication is that any insight that the analyst has concerning the physical nature of the system cannot be
readily utilized in subsequent analyses using the simplified model.
An application of frequency weighted internally balanced model reduction was presented in Schmidt and
Newman (1989). This study demonstrated the importance of appropriate model reduction for application in
control synthesis. If the reduced order model does not show good agreement with the full order model in the
region of crossover, even where the transfer functions have relatively small magnitude, the resulting control
law may not perform as expected when applied to the full order model (even to the point of destabilizing the
system).
12
Modeling and Model Simplification of Aeroelastic Vehicles
Literal Simplification
The last model reduction method which will be discussed here and one which is often overlooked is
literal approximation. This method is based on first-order perturbation theory and can, in principle, be
applied to high order models.
An advantage of this approach is that it allows one to identify the cause and effect relationships between
physical parameters and dynamic behavior. A disadvantage of this approach is that it is tedious to apply to
more than very simple systems. The recent advances in symbolic mathematics computer programs however
have fostered a renewed interest in this approach [Schmidt and Newman (1988)].
In an earlier section the equations of motion for an elastic aircraft were described in a literal form using
modal structural representations. This form of the equations of motion lends itself to literal (symbolic)
formulation of system transfer functions. This can be accomplished by hand or with the aid of one of the
many symbolic mathematics computer programs.
Consider literal representations of the numerator and denominator polynomials for a pitch-rate-to-
elevator transfer function of an elastic airplane in which the short period approximation has been applied
[McRuer, Ashkenas, and Graham (1973)] and with one structural mode included in the model. These
polynomials are presented in Table 4.
The parameters that appear in the polynomials represent stability and control derivatives (Z, M, and F)1,
structural parameters (co, _, and _),)2, and the flight speed V. Those terms with subscripts o_ and q are
associated with the rigid-body degrees of freedom (angle of attack and pitch rate, respectively). Those terms
with the subscript 5 e are associated with the elevator deflection. Those terms with subscripts 1"1and _1 are
associated with the modeled elastic degree of freedom and its time derivative.
Table 4 Example of literal transfer function polynomials - G_N(s)
" - D(s)
N(s)
D(s)
Zso, r. . 2 (2_co- Ffi)s + (co2 - t_'s[M_FqS +Fa(s 2 -MqS)] }--q-/stM_ks + -F.q)) + Fcz(M/ls + Mrl)] +
Zct 2+ (2;co- 1]l)s + (co2 - Frl))- F_('v-Zqr'__ Z--R Zcts+ V)]-_)'S[FqS(S--_-) +Fct(1 + --_)s] } +M_ {s[(s--v-)(s
Zo_ Z:. Z Zct 2F8 {s[(s--_-)(M/ls+M.q)+Mt_(-_!s+_)]- @'s[(s--_)(s -MqS)-Mct(1 +-_-q)s] }
Z_ ZcZ/ls + Zn)[M F s + F=(s2-MqS)] - (M/lS + M_I)[FqS(S--_) + F_(1 +-_q )s] +--k'_- V _ q
Za 2 Z(s 2 + (2_co- Ffi)s + (co2-Fn))[(s-_)(s -MqS)-Ma( 1 +-_)s]
1 Z is the force oriented along the body axis orthogonal to the plane of the wing (and is predominantly lift), M is the pitching
moment, and F is the generalized force associated with the elastic degree of freedom.
2 to and _ are the invacuo frequency and damping of the elastic mode, and _' is the mode slope at the point where pitch rate is
measured.
13
M.R. Waszak and C.S. Buttrill
The numerical form of the transfer function structure represented by the polynomials in Table 4 is
presented in Eqn. (9) for a high speed transport aircraft example from Waszak and Schmidt (1988). The
numerical form of the transfer function was obtained by truncating the forward velocity perturbation degree
of freedom and residualizing the second, third and fourth structural modes from the elastic equations of
motion of the aircraft. Note that the numerator has three real roots and one root at the origin and the
denominator has two pairs of complex conjugate roots and one root at the origin.
G_qe= s(-_+ _ _s_s+---_5.51)13"06s(s+ 0.231)(s- 3.362)(s + 3.959)(9)
Once the numerator and denominator polynomials of the desired transfer function are obtained, the
approximate terms are chosen so that the following two criteria are satisfied.
a) The literal expressions must factor into the same form as the original polynomials (e.g. order of
polynomial, number of real and complex roots), and
b) numerical values based on the simplified terms should accurately approximate the values based on
the original polynomials.
The underlined terms of the numerator and denominator polynomials in Table 4 involve the key model
parameters (stability derivatives and structural parameters), which factor into the appropriate form, and
result in approximate polynomials with the desired properties described above. These terms are used to
obtain the approximate literal model presented in Table 5. The corresponding numerical values for this
approximate model are also presented in Eqn. (10).
_qe = 13.06s(s + 0.416)(s- 3.265)(s + 4.177)s(s2+ 1.246s + 3.758)(s2+ 0.62 ls + 34.83)
(10)
Table 5 Approximate literal transfer function polynomials _q_e /q(s)
6(s)
Zet - (b22_--4c) 1/2 + (b2.2-- 4c) 1/2(Mao-CFao)s[s+(-ff)][s +(b ._ .)][s+(b .)]
Za Zs[s 2 + (___E__ Mq)S + (_-- Mq- (1 +_q)Ma)] (s 2 + (2-_0- F./1)s+ (032-Frl))
b = [Ms (2_¢-o-Frl) + _'F_iM q ] / (M_ - q_'F_i ) c = M_io(m2 -Frl ) / (M_- qb'Fa )
14
Modeling and Model Simplification of Aeroelastic Vehicles
Note that the numerical values from the approximate model agree to varying degree with the "truth" model
in Eqn. (9). Those terms which are deemed to be of insufficient accuracy can be modified by computing
correction terms.
Corrections to the approximate factors can be obtained in literal form by applying perturbation theory.
Expanding the true polynomial coefficient, Pi, in a Taylor series about the approximate value, l_i, allows one
to compute literal corrections. This requires that the Taylor series be truncated after the first-order term,
Pi --" Pi + _ AZZ
(11)
Here % is the vector of model parameters that contribute to the value of the polynomial coefficient Pi" The
correction A_ clearly requires literal expressions for Pi - Pi and _Pi-- tO be available. The difference
expression, Pi - Pi, is simply what remains after the approximate factor is extracted from the literal
expression for Pi and corresponds to the non-underlined terms in Table 4. The partial derivative term can be
obtained by direct symbolic differentiation with respect to the model parameters, _.
The correction factors can be used either to enhance the accuracy of the approximate model or to identify
the sensitivity of the simplified model to variations in various physical parameters.
A numerical form of the literally simplified model can be obtained by substituting the values of the
various parameters directly into the literal expressions. The literally simplified model, unlike strictly
numerical models, can be used to assess the reason behind mismatches with the original model. If an error
occurs at a particular frequency, the model parameters which are dominant at that frequency and contribute
significantly to the error can be directly identified. In addition, the impact of potential variations or
uncertainties in a particular model parameter can be quantified in terms of its effect on the vehicle response.
An example of literal model simplification is presented in Schmidt and Newman (1988). This approach
was shown to yield excellent results when applied to a high speed transport aircraft. Furthermore, the
closed-form analytical expressions for the key dynamic characteristics that result allow one to identify
critical parameters affecting the vehicle dynamics.
Summary
Each of the model reduction methods described here have clear advantages and disadvantages. As such,
it is unlikely that any one method will be able to satisfy all model simplification needs. In fact, the analyst
should make efforts to recognize the strengths and weaknesses of each method and use one which best suits
the particular needs.
These methods are not necessarily mutually exclusive either. One method can be used to compliment
another and enhance ones understanding of the vehicle's dynamic behavior. For example, truncation and
residualization may be used initially to reduce the model to a tractable form. Then literal methods may be
used to identify the sensitivity of the model to parameter variations and uncertainties. Finally, internally
15
M.R. Waszak and C.S. Buttrill
balanced reduction might be used to obtain a numerical form of the model or further simplify a numerical
version of the literal model.
The most important recommendation, however, is to use caution whenever applying model
simplification to aeroelastic systems. Blindly applying any simplification method will lead to a simpler
model, but one which may not accurately convey the important dynamic characteristics which influence the
vehicle behavior.
Concluding Remarks
The objective of this paper was to emphasize some of the key issues associated with modeling elastic
aircraft for dynamic analysis and control law synthesis. Emphasis has been placed on the importance of
initially developing high fidelity models which are subsequently simplified for particular applications. This
approach assures that the salient features of the vehicle dynamics will be represented in the design model. In
addition, this approach results in a model structure which is consistent and applicable over the entire
development cycle, including preliminary design: This is especially important in allowing control
technologies to play a role in shaping the vehicle configuration.
The development of two modeling approaches were specifically addressed with particular attention paid
to the underlying assumptions. The first approach results in a model structure with which literal models can
be developed. The second modeling approach addressed the issues associated with including additional
inertial coupling terms in the model and provided guidelines for when inertial coupling should be included.
The importance of model simplification was also addressed by considering the advantages and
disadvantages of four model simplification methods. The first two simplification methods, truncation and
residualization, represent traditional approaches. These were viewed in a way which resulted in some
guidelines for when they can be legitimately applied. The third method, internally balanced reduction,
represents the newer model simplification approaches which provide added capabilities subject to certain
limitations which were discussed. The last method, literal simplification, summarized an approach which,
while currently often overlooked, will become more attractive as symbolic mathematics computer programs
become more capable.
The results from the studies described herein and the perceived need for accurate models of elastic
aircraft for control design applications indicate that more emphasis should be placed on the modeling
process. It is recommended that model development should involve both formulating equations of motion
and model simplification. Each phase should be treated separately but with knowledge of the other. This
approach makes more likely the possibility that the salient aspects of the system dynamics will be accurately
modeled.
16
Modeling and Model Simplification of Aeroelastic Vehicles
References
Bacon, B.J. and Schmidt, D.K., "Multivariable Frequency-Weighted Order Reduction," Journal of
Guidance, Control and Dynamics, Vol. 12, No. 1, Jan-Feb, 1989, pp. 97-107.
Brogan, W.L., Modern Control Theory, Quantum Publishers, Inc., New York, 1974.
Buttrill, C.S., Zeiler, T.A., and Arbuckle, P.D., "Nonlinear Simulation of a Flexible Aircraft in Maneuvering
Flight," AIAA Paper 87-2501-CP, AIAA Flight Simulation Technologies Conference, Monterey,
CA, August, 1987.
Canavin, J.R. and Likins, P.W., "Floating Reference Frames for Flexible Spacecraft," Journal of Spacecraft
and Rockets, Vol. 14, NO. 12, Dec. 1977, pp 724-732.
Cerra, J.J., and Noll, T.E., "Modelling of Rigid-Body and Elastic Aircraft Dynamics for Flight Control
Development," AIAA Paper Number 86-2232, 1986.
D'Azzo, J.J., and Hoopis, C.H., Linear Control System Analysis and Design: Conventional and Modern,
McGraw-Hill, Inc., New York, 1975.
Enns, D.F., "Model Reduction for Control System Design," Ph.D. Dissertation, Dept. of Aeronautics and
Astronautics, Stanford University, Stanford, CA, June, 1984.
Glover, K., "All Optimal Hankel-Norm Approximations of Linear Multivariable Systems and Their L °°-
Error Bounds," International Journal of Control, Vol. 39, No. 6, pp. 1115-1193, 1984.
Kokotovich, P.V., O'Malley, R.E., and Sannuti, P., "Singular Perturbations and Order Reduction in Control
Theory - An Overview," Automatica, Vol. 12, 1976, pp. 123-132.
McRuer, D., Ashkenas, I., and Graham, D., Aircraft Dynamics and Automatic Control, Princeton University
Press, Princeton, New Jersey, 1973.
Milne, R.D., "Dynamics of the Deformable Aeroplane (Part I & 11)," Queen Mary College, University of
London, Reports and Memoranda No. 3345, September 1962.
Milne, R.D., "Some Remarks on the Dynamics of Deformable Bodies," AIAA Journal, Vol. 6, March 1968,
pp. 556-558.
Roger, K.L., "Airplane Math Modeling Methods for Active Control Design," Structural Aspects of Active
Control, AGARD-CP-228, April 1977.
Schwanz, R.C., "Consistency in Aircraft Structural and Flight Control Analysis," Structural Aspects of
Active Control, AGARD-CP-228, April, 1977.
Schmidt, D.K., "Pilot Modeling and Closed-Loop Analysis of Flexible Aircraft in the Pitch Tracking Task,"
Journal of Guidance, Control and Dynamics, Vol. 8, No. 1, Jan.-Feb. 1985, pp. 56-61.
Schmidt, D.K. and Newman, B., "Modeling, Model Simplification and Stability Robustness with Aeroelastic
Vehicles," AIAA Paper No. 88-4079-CP, AIAA Guidance, Navigation, and Control Conference,
Minneapolis, MN, August, 1988.
17
M_R. Waszak and C.S. Buttrill
Schmidt, D.K. and Newman, B., "On the Control of Elastic Vehicles - Model Simplification and Stability
Robustness," AIAA Paper No. 89-3558, AIAA Guidance, Navigation, and Control Conference,
Boston, MA, August, 1989.
Strang, G., Linear Algebra andlts Applications, Academic Press, Inc., New York, 1980.
Swaim, R.L. and Poopaka, S., "An Analytical Pilot Rating Method for Highly Elastic Aircraft," Journal of
Guidance, Navigation and Control, Vol. 5, No. 6, Nov.-Dec. 1982, pp. 578-582.
Waszak, M.R., Davidson, J.B., and Schmidt, D.K., "A Simulation Study of the Flight Dynamics of Elastic
Aircraft: Volume 1 - Experiment, Results and Analysis," NASA CP 4102, Dec. 1987.
Waszak, M.R. and Schmidt, D.K., "Flight Dynamics of Aeroelastic Vehicles," Journal of Aircraft, Vol. 25,
No. 6, June 1988, pp. 263-271.
Zeiler, T.A. and Buttrill, C.S., "Dynamic Analysis of an Unrestrained, Rotating Structure Through
Nonlinear Simulation," AIAA Paper 88-2232-CP, Proceedings of the AIAA/ASME/ASCE/AHS 27th
Structures, Structural Dynamics and Materials Conference, Williamsburg, VA, April, 1988, pp 167-
174.
18
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September 1992 Technical Memorandum
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Modeling and Model Simplification of Aeroelastic Vehicles: An Overview 505-64-52-03
6. AUTHOR(S)
Martin R. Waszak, Carey S. Buttrill, and David ]C Schmidt
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
NASA Langely Research CenterHampton, VA 23681-0001
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National Aeronautics and Space Administration
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Waszak and Buttrill: Langley Research Center, Hampton, VA; and Schmidt: Arizona State University, Tempe, Arizona
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13. ABSTRACT (Maximum 200 words)
The rigid-body degrees of freedom and elastic degrees of freedom of aeroelastic vehicles are typically treated
separately in dynamic analysis. Such a decoupling, however, is not always justified and modeling assumptions
that imply decoupling must be used with caution. The frequency separation between the rigid-body and elastic
degrees of freedom for advanced aircraft may no longer be sufficient to permit the typical treatment of the vehicle
dynamics. Integrated, elastic vehicle models must be developed initially and simplified in a manner appropriateto and consistent with the intended application. This paper summarizes key results from past research aimed at
developing and implementing integrated aeroelastic vehicle models for flight controls analysis and design. Three
major areas will be addressed; 1) the accurate representation of the dynamics of aeroelastic vehicles, 2)
properties of several model simplification methods and 3) the importance of understanding the physics of thesystem being modeled and of having a model which exposes the underlying physical causes for critical dynamic
characteristics.
14. SUBJECT TERMS
dynamics and control, mathematical models, model simplification, aeroelasticity
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